1. **State the problem:** Simplify the expression $$\sqrt[3]{\frac{49}{27}} \cdot j^{\frac{1}{3}}$$ where $j$ is a variable. 2. **Recall the cube root and exponent rules:** - The cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator: $$\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$$ - Multiplying expressions with the same root and exponent can be combined: $$\sqrt[3]{x} \cdot y^{\frac{1}{3}} = (x \cdot y)^{\frac{1}{3}}$$ 3. **Apply the cube root to the fraction:** $$\sqrt[3]{\frac{49}{27}} = \frac{\sqrt[3]{49}}{\sqrt[3]{27}}$$ 4. **Simplify the denominator:** $$\sqrt[3]{27} = 3$$ because $3^3 = 27$. 5. **Rewrite the expression:** $$\frac{\sqrt[3]{49}}{3} \cdot j^{\frac{1}{3}} = \frac{\sqrt[3]{49} \cdot j^{\frac{1}{3}}}{3}$$ 6. **Combine the cube roots:** $$\frac{\sqrt[3]{49} \cdot j^{\frac{1}{3}}}{3} = \frac{\sqrt[3]{49 \cdot j}}{3}$$ 7. **Final simplified form:** $$\boxed{\frac{\sqrt[3]{49j}}{3}}$$ This is the simplest exact form unless you want to approximate the cube root of 49.